The behavior of a particle moving according to the laws of quantum mechanics in a field, which potential changes over time, is studied. The method of unitary transformations for solving the temporal Schrödinger equation is considered. The reduced Hamiltonian of the system of a quantum-mechanical particle in time-dependent potential is obtained, as well as the total operator of the evolution of such a system. We find a new version of the unitary transformation, which, compared to the known ones, simplifies solving the problem in analytical form. This transformation associates the linear potential quantum model with time-dependent parameters and freeparticle system. Based on the constructed evolution operator, we consider the wave packet dispersion. It is shown that the wave packet dispersion for this model is the same as that for a free particle. This conclusion is the general result of the behavior of a quantummechanical particle moving in the field of a timedependent linear potential.The behavior of a particle moving according to the laws of quantum mechanics in a field, which potential changes over time, is studied. The method of unitary transformations for solving the temporal Schrödinger equation is considered. The reduced Hamiltonian of the system of a quantum-mechanical particle in time-dependent potential is obtained, as well as the total operator of the evolution of such a system. We find a new version of the unitary transformation, which, compared to the known ones, simplifies solving the problem in analytical form. This transformation associates the linear potential quantum model with time-dependent parameters and freeparticle system. Based on the constructed evolution operator, we consider the wave packet dispersion. It is shown that the wave packet dispersion for this model is the same as that for a free particle. This conclusion is the general result of the behavior of a quantummechanical particle moving in the field of a timedependent linear potential.

Journal of Physics and ElectronicsDec 30, 20220 citations

Deformation quantization of linear dissipative systems

V. G. Kupriyanov, S. L. Lyakhovich, A. A. Sharapov

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\\partial_t+...$ , differentiating the $\\ast_t$ -product. As in the usual case, the $\\ast_t$ -algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\\mathrm{Tr}_t$ . Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\\partial_t+...$, differentiating the $\\ast_t$-product. As in the usual case, the $\\ast_t$-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\\mathrm{Tr}_t$. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.

Jan 1, 20057 citations

Time-Dependent Quantum Mechanics

Zoltán Papp

Springer eBooksJan 1, 20260 citations

Deformation quantization of linear dissipative systems

V G Kupriyanov, S L Lyakhovich, A A Sharapov

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation, we develop a consistent deformation quantization procedure involving a non-stationary star-product ∗ t and an 'extended' operator of time derivative D t = ∂ t + ⋅⋅⋅, differentiating the ∗ t -product. As in the usual case, the ∗ t -algebra of physical observables is shown to admit an essentially unique (time-dependent) trace functional Tr t . Using these ingredients, we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of a non-stationary Poisson bracket, i.e. the corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation, we develop a consistent deformation quantization procedure involving a non-stationary star-product ∗ t and an 'extended' operator of time derivative D t = ∂ t + ⋅⋅⋅, differentiating the ∗ t -product. As in the usual case, the ∗ t -algebra of physical observables is shown to admit an essentially unique (time-dependent) trace functional Tr t . Using these ingredients, we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.

arXiv:quant-ph/0505023Journal of Physics A Mathematical and GeneralAug 31, 200519 citations

*_t

and an ``extended'' operator of time derivative

D_t=\\partial_t+...

, differentiating the

\\ast_t

-product. As in the usual case, the

\\ast_t

-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional

\\mathrm{Tr}_t

. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.